Problems

Two opposite corners were removed from an $8\times 8$ chessboard. Can you cover this chessboard with $1\times 2$ rectangular blocks?

The numbers 1 through 12 are written on a board. You can erase any two of these numbers (call them $a$ and $b$) and replace them with the number $a+b-1$. After 11 such operations, there will be just one number left. What could this number be?

If a magician puts 1 dove into his hat, he pulls out 2 rabbits and 2 flowers from it. If the magician puts 1 rabbit in, he pulls out 2 flowers and 2 doves. If he puts 1 flower in, he pulls out 1 rabbit and 3 doves. The magician starts with 1 rabbit. Could he end up with the same number of rabbits, doves, and flowers after performing his hat trick several times?

One small square of a $10\times 10$ square was removed. Can you cover the rest of it with 3-square $L$-shaped blocks?

There are real numbers written on each field of a $m\times n$ chessboard. Some of them are negative, some are positive. In one move we can multiply all the numbers in one column or row by $-1$. Is that always possible to obtain a chessboard where sums of numbers in each row and column are nonnegative?

A $7\times 7$ square was tiled using $1\times 3$ rectangular blocks. One of the squares has not been covered. Which one can it be?

Tom found a large, old clock face and put 12 sweets on the number 12. Then he started to play a game with himself. In each move he moves one sweet to the next number clockwise, and some other to the next number anticlockwise. Is it possible that after finite number of steps there is exactly 1 of the sweets on each number?

Can you cover a $13\times 13$ square using two types of blocks: $2\times 2$ squares and $3\times 3$ squares?

What time is it going to be in $2019$ hours from now?

What is the remainder of $1203\times 1203-1202\times 1205$ when divided by $12$?